Thanks a lot for this reply. I tried a proof along very similar lines. But I don't know why the set of ordered pairs (a,b) such that there exists a disc D whose boundary circle meets the Jordan polygon at gamma(a) and gamma(b) is a closed subset of S^1 x S^1.

I know why it's a subset of S^1 x S^1 but I don't know why it's a closed subset.